Perelman's l-distrance

This talk is a preparation of the necessary tools for proving the non-collapsing

results. The L-length defined by Perelman is the analog of an energy path, but

defined in a Riemannian manifold context. The length is used to define the l

reduced distance and later on, the reduced volume. So far the properties of the

l-length have two applications in the proof of the Poincar´e conjecture. Associated

with the notion of reduced volume, they are used to prove non-collapsing results

and also to study the κ- solutions.